Central Binomial Sums, Multiple Clausen Values and Zeta Values
J. M. Borwein, D. J. Broadhurst, J. Kamnitzer
TL;DR
The paper establishes deep connections between central binomial sums, alternating Apéry sums, and a new class of polylogarithmic values called multiple Clausen values (MCVs) evaluated at the sixth root of unity. It develops a comprehensive framework for MCVs, including their real/imaginary decompositions (mgl/mcl), duality relations, and generating-function methods that express central binomial sums in terms of MCVs and zeta-values, supported by explicit low-weight evaluations. It also proposes dimensional conjectures for MCV spaces and presents a golden-ratio polylogarithmic ladder for Apéry-type sums, combining exact results and extensive empirical relations to reveal the rich structure of these constants and their interconnections. Together, these results unify non-alternating and alternating binomial sums within a polylogarithmic framework with potential implications for the irrationality of odd zeta-values and quantum-field-theory related constants. The work provides both rigorous identities and computational strategies (including PSLQ-based checks) and outlines directions for further structural understanding of MCVs.
Abstract
We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Apéry sums). The study of non-alternating sums leads to an investigation of different types of sums which we call multiple Clausen values. The study of alternating sums leads to a tower of experimental results involving polylogarithms in the golden ratio. In the non-alternating case, there is a strong connection to polylogarithms of the sixth root of unity, encountered in the 3-loop Feynman diagrams of {\tt hep-th/9803091} and subsequently in hep-ph/9910223, hep-ph/9910224, cond-mat/9911452 and hep-th/0004010.